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{{Short description|Generating pseudo-random numbers that follow a probability distribution}} ...'' is the [[Numerical analysis|numerical]] practice of generating [[pseudo-random number]]s (PRN) that follow a given [[probability distribution]]. ...

...cial cases of the class of [[ratio distribution]]s, in which the numerator random variable has a [[degenerate distribution]]. In general, given the [[probability distribution]] of a random variable ''X'' with strictly positive support, it is possible to find the ...

...isfies the conditions of [[Banach fixed-point theorem]] (the domain is a [[non-empty]] [[complete metric space]]) then it is sufficient to prove convergen ====Non-expansion mappings==== ...

...s to create a bounded set, which can then be sampled uniformly to generate random variables following the original distribution. One feature of this method i ...an define a bounding box around it (a uniform proposal distribution), draw uniform samples in the box and return only the x coordinates of the points that fal ...

...d not be predictable, but even more, it should be indistinguishable from a random string. Let <math> \mathcal{A}^f </math> denote the algorithm <math> \mathc ...by <math> \mathcal{I}\mathcal{G} (n) </math>, instance generator, and the random choice of the function <math> R_{a}(x) </math> among the set of all <math> ...

...hat has been proven to produce truly random output from any type of weakly random source. ...iased" source and output a distribution that appears unbiased. The weakly random source will always be longer than the extractor's output, but an efficient ...

...irected graph {{mvar|G}} is the smallest number {{mvar|d}} such that every non-empty subgraph of {{mvar|G}} has at least one vertex of degree at most {{mv ...by repeatedly choosing a uniformly random vertex and choosing uniformly at random among all the available colors for that vertex, including the color it alre ...

...o [[encoding|encode]] each possible [[permutation]] of a sequence of ''n'' numbers. It is an instance of a scheme for [[Permutation#Numbering permutations|num ...mber for which only a single fixed value is allowed; ''every'' sequence of numbers chosen from these sets encodes a single permutation. While several [[encodi ...

...on (mathematics)|reflected]] around a vertical line at some value of the [[random variable]] represented by the distribution. This vertical line is the line :<math> f(x_0-\delta) = f(x_0+\delta) </math> for all real numbers <math>\delta ,</math> ...

...abius function.png|thumb|Extension of the function to the nonnegative real numbers.]] ...ndependent]] [[uniform distribution (continuous)|uniformly distributed]] [[random variable]]s on the [[unit interval]]. That distribution has an expectation ...

...zero probability all of these events can be avoided. However, the lemma is non-constructive in that it does not provide any insight on ''how'' to avoid th ...b>n</sub>''} are determined by a finite collection of mutually independent random variables, a simple [[Las Vegas algorithm]] with [[ZPP (complexity)|expecte ...

...rwise, if the condition is left random, the result of conditioning is also random. Example: A fair coin is tossed 10 times; the [[random variable]] ''X'' is the number of heads in these 10 tosses, and ''Y'' is th ...

|title=Efficient Multiply-with-Carry Random Number Generators with Maximal Period ...tate change an FCSR generates an infinite, eventually periodic sequence of numbers in <math>S</math>. ...

...e.g. the stadium-shaped cavity on the right) tend to become gradually more uniform with increasing mode number.]] ...nian mechanics|Hamiltonian]] tend to a [[Uniform distribution (continuous)|uniform distribution]] in the classical [[phase space]]. This is consistent with t ...

...ution, with (some of) the parameters of that distribution themselves being random variables. ...[[marginal distribution|marginalizing]] (integrating) over the ''latent'' random variable(s) representing the parameter(s) of the parametrized distribution ...

...scribe the lengths of [[longest common subsequence]]s of [[random variable|random]] [[string (computer science)|strings]]. Although the existence of these co | title = Longest common subsequences of two random sequences ...

...the concept of [[risk measure]] is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, e ...obability measure]] on <math>\mathcal{F}</math>. Let <math>X</math> be a [[random variable]] and <math>\mathbf{L}_{M^+}</math> be the set of all [[Borel meas ...

...]], a '''subgaussian distribution''', the distribution of a '''subgaussian random variable''', is a [[probability distribution]] with strong tail decay. More Often in analysis, we divide an object (such as a random variable) into two parts, a central bulk and a distant tail, then analyze e ...

===Real numbers=== ...real number can be generated with near certainty by sampling each digit at random in an infinite unending process.</ref> Note that the programs that act on t ...

{{Short description|Mathematical inequality explaining concentration of random variables}} ...when the CDF is difficult to calculate or even the exact form of the first random variable is unknown that the applicable concentration inequalities provide ...